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arXiv · 1803.05628

The total zero-divisor graph of commutative rings

Alen Đurić·Sara Jevđenić·Polona Oblak·Nik Stopar
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Abstract

In this paper we initiate the study of the total zero-divisor graphs over commutative rings with unity. These graphs are constructed by both relations that arise from the zero-divisor graph and from the total graph of a ring. We characterize Artinian rings with the connected total zero-divisor graphs and give their diameters. Moreover, we compute major characteristics of the total zero-divisor graphs of the ring ${\mathbb Z}_m$ of integers modulo $m$ and prove that the total zero-divisor graphs of ${\mathbb Z}_m$ and ${\mathbb Z}_n$ are isomorphic if and only if $m=n$.

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Alen Đurić, Sara Jevđenić, Polona Oblak, Nik Stopar. 2018-03-15. The total zero-divisor graph of commutative rings. https://doi.org/10.1142/s0219498819501901

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